Ince equation

In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation

: $w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. \,$

When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when $p=1, \eta\pm\xi=1$ , then it has a closed-form solution{{cite journal |last1=Cheung |first1=Tsz Yung |title=Liouvillian solutions of Whittaker-Ince equation |journal=Journal of Symbolic Computation |volume=115 |issue=March-April 2023 |pages=18-38 |doi=10.1016/j.jsc.2022.07.002}}

$w(z)=Ce^{-iz}(e^{2iz}\mp 1)$

where $C$ is a constant.

References

{{Citation | last1=Boyer | first1=C. P. | last2=Kalnins | first2=E. G. | last3=Miller | first3=W. Jr. | title=Lie theory and separation of variables. VII. The harmonic oscillator in elliptic coordinates and Ince polynomials |mr=0372384 | year=1975 | journal=Journal of Mathematical Physics | issn=0022-2488 | volume=16 | issue=3 | pages=512–517|bibcode = 1975JMP....16..512B |doi = 10.1063/1.522574 | url=http://researchcommons.waikato.ac.nz/bitstream/10289/1243/1/Kalnins%20variables%207.pdf | hdl=10289/1243 | hdl-access=free }}
{{Citation | last1=Magnus | first1=Wilhelm | author1-link=Wilhelm Magnus | last2=Winkler | first2=Stanley | title=Hill's equation | url=https://books.google.com/books?id=ML5wm-T4RVQC | publisher=Interscience Publishers John Wiley & Sons\, New York-London-Sydney | series=Interscience Tracts in Pure and Applied Mathematics, No. 20 | isbn=978-0-486-49565-1 |mr=0197830 | year=1966}}
{{Citation | last1=Mennicken | first1=Reinhard | title=On Ince's equation | publisher=Springer Berlin / Heidelberg |mr=0223636 | year=1968 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | volume=29 | issue=2 | pages=144–160 | doi=10.1007/BF00281363 | bibcode=1968ArRMA..29..144M| s2cid=122886716 }}
{{dlmf|id=28.31|title=Equations of Whittaker–Hill and Ince|first=G. |last=Wolf}}

Category:Ordinary differential equations

Ince equation

See also

References